Multiple sparse component analysis based on subspace selective search algorithm

Abstract

Sparse component analysis (SCA) is an approach for linear matrix factorization in an instantaneous mixing system when the number of sensors is fewer than the number of sources. SCA assumes that matrix source (S) contains as many zeros as possible. According to the Georgiev's proof, under some nonstrict conditions on sparsity of the sources, called k-sparse component analysis (k-SCA), we are able to estimate both mixing system (A) and sparse sources (S) uniquely. This paper studies the problem of underdetermind blind identification (UBI) in order to estimate the mixing matrix A based on subspace clustering scheme with k-SCA assumptions. Most k-SCA based algorithms have been designed when there are only at most k = m − 1 active sources in each time instant, where m is the number of sensors. First, we address the issue of multiple active sources i.e. when there are L active sources (0 ≤ L ≤ m − 1). Second, we propose an algorithm for joint or continuous subspace clustering and estimating of the channels in order to design an online scenario to estimate the mixing matrix columns and detect the number of sources as the mixture vectors are received sequentially. Third, our proposed algorithm "subspace selective search" (S 3 ) deals with the outliers within subspace clustering process and improve the accuracy of channel estimation. Numerical simulations are reported to confirm the advantages of our UBI method.

A Novel Underdetermined Source Recovery Algorithm Based on k-Sparse Component Analysis

“…Matrix A is assumed to be known or could be identified using the existing UBI algorithms [23][24][25][26][27][28][29][30]. Note that our algorithm focuses on the source recovery but we add Gaussian noise to A with a predefined N M SE A in order to evaluate the sensitivity of algorithms to any error in estimation A:…”

“…Based on Georgive et al 's proof [4], as long as k ≤ m − 1, where k is the number of active sources at each column of S, we can identify A and estimate S. Essentially, Georgive's proof guarantees both stages of UBSS (UBI and USR), however he and his co-authors did not provide any detailed or robust algorithm to justify the stages for UBI and USR. Albeit, few methods, including two approaches proposed by our group recently, extended this method to solve the UBI problem [23][24][25][26][27][28][29][30].…”

“…Therefore, a dedicated method for solving the above-mentioned USR problem is necessary especially for applications with more dense sparse sources. In this regard, we have tried to introduce a fundamental approach when k is known or detected using the proposed methods in [27,30] for each time instant.…”

“…During these years, some k-SCA based UBSS methods have been proposed based on the k-SCA assumptions in the literature. Most of the developed k-SCA algorithms for solving UBSS problem focus on identifying the mixing matrix (UBI) [23][24][25][26][27][28][29][30] and utilise conventional source recovery algorithms i.e. greedy methods, e.g., orthogonal matching pursuit (OMP) [31][32][33], relaxation methods, e.g., basis pursuit (BP) [11,[34][35][36] and smoothed L0 (SL0) algorithm [37,38].…”

“…The proposed method is not based on any BP algorithms and its approach is more similar to the greedy based source recovery approaches. Indeed, based on the k-SCA assumptions, we propose a novel and effective method to solve the USR problem assuming the mixing matrix (A) is known or could be identified using methods such as those proposed in [23][24][25][26][27][28][29][30]. It detects the generating subspaces of each mixed data based on eigenvalue decomposition of concatenated mixed data and the columns of the mixing matrix.…”